Scalar control actions \(u_1(t),\ldots , u_{N}(t)\)  with \(N\)  to be determined later, act on the first equation and parameter \(\theta \)  taking values in \(\lbrace 0,1\rbrace \)  determines whether the control is placed internally (\(\theta =1\) ) or on the right boundary (\(\theta =0\) ). Functions \(b_1(x),\cdots ,b_{N}(x)\)  in \(L^2(0,L)\)  describe how the internal control actions are distributed in \([0,L]\)  and are subject to some constraints given below. On the boundaries, we have \(\gamma _{ij}\in \mathbb {R}\)  satisfying \(\gamma _{i1}^2+\gamma _{i2}^2\ne 0,\quad i=1,2\)  when \(\theta =1\) , whereas for \(\theta =0\) , we have the additional restriction that \(\gamma _{12}=1-\gamma _{11}\)  and \(\gamma _{22}=1-\gamma _{21}\)  with \(\gamma _{11}, \gamma _{21} \in \lbrace 0,1\rbrace \) . Condition on \(\gamma _{ij}\)  restricts the type of boundary conditions to either Neumann or Dirichlet ones for the boundary control case, whereas for the distributed control case, we can have more general boundary conditions of mixed type. For boundary control, this stands as a sufficient condition for invertibility of a matrix which leads to stabilizability as it is revealed later.
Systems of the form () can model for instance Turing instability (see [1]}) and instability of slime mold amoebae aggregation (see [2]}).
We make the following assumption:
